Is the sum of digits of $3^{1000}$ a multiple of $7$?

The sum of the digits of $3^{1000}$ can be computed using a computer. It is equal to $2142$, so the answer is positive.

**Is there a short proof that the sum of the digits of $3^{1000}$ is a multiple of $7$ without using a computer?**

Do you have any advice to solve this type of problem (without programming of course!)?

The results below are known:

$3^{1000}$ has $478$ digits, and so the sum is at most $4302$ ($9\cdot478$).

This sum is a multiple of $9$.

The last four digits of $3^{1000}$ are $0001$.

Context: We are a group of 3 French people working on it since 2007. It's a little exercise I found in my high school book (printed in 2007) which is pretty complicated. The one who created this exercise doesn't know the answer.

This question was previously asked on Math.SE (link).